{
  "id": "2308.11586",
  "version": "v1",
  "published": "2023-08-22T17:34:55.000Z",
  "updated": "2023-08-22T17:34:55.000Z",
  "title": "Uniqueness of algebra norm on quotients of the algebra of bounded operators on a Banach space",
  "authors": [
    "Max Arnott",
    "Niels Jakob Laustsen"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "We show that for each of the following Banach spaces~$X$, the quotient algebra $\\mathscr{B}(X)/\\mathscr{I}$ has a unique algebra norm for every closed ideal $\\mathscr{I}$ of $\\mathscr{B}(X)\\colon$ - $X= \\bigl(\\bigoplus_{n\\in\\N}\\ell_2^n\\bigr)_{c_0}$\\quad and its dual,\\quad $X= \\bigl(\\bigoplus_{n\\in\\N}\\ell_2^n\\bigr)_{\\ell_1}$, - $X= \\bigl(\\bigoplus_{n\\in\\N}\\ell_2^n\\bigr)_{c_0}\\oplus c_0(\\Gamma)$\\quad and its dual, \\quad $X= \\bigl(\\bigoplus_{n\\in\\N}\\ell_2^n\\bigr)_{\\ell_1}\\oplus\\ell_1(\\Gamma)$,\\quad for an uncountable cardinal number~$\\Gamma$, - $X = C_0(K_{\\mathcal{A}})$, the Banach space of continuous functions vanishing at infinity on the locally compact Mr\\'{o}wka space~$K_{\\mathcal{A}}$ induced by an uncountable, almost disjoint family~$\\mathcal{A}$ of infinite subsets of~$\\mathbb{N}$, constructed such that $C_0(K_{\\mathcal{A}})$ admits \"few operators\". Equivalently, this result states that every homomorphism from~$\\mathscr{B}(X)$ into a Banach algebra is continuous and has closed range. The key step in our proof is to show that the identity operator on a suitably chosen Banach space factors through every operator in $\\mathscr{B}(X)\\setminus\\mathscr{I}$ with control over the norms of the operators used in the factorization. These quantitative factorization results may be of independent interest.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-22T17:34:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "bounded operators",
      "uniqueness",
      "suitably chosen banach space factors",
      "unique algebra norm",
      "quotient algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}